On the Acoustical Radiation of an Emitter Vibrating Freely or in a Wall of Finite Dimensions

An acoustical radiation field excited by an emitter vibrating freely or in a wall of finite dimensions is considered as superposed by two fields. The first of these is that one where the same emitter is vibrating in an infinite wall and the second is that which (i) causes the resultant field in the free part of the plane going through the wall to vanish, and (ii) has a normal derivative which vanishes on the surface of the emitter and of the wall. While the first field may be considered as known from other papers, the second is computed from an integro-differential equation that follows from Rayleigh's formula. The equations expressing the velocity potential distribution and those deduced from it are written in an abstract form by means of the Dirac bra-vectors, ket-vectors, and linear operators represented by the corresponding matrices. This method of solving the given special diffraction problem of a scalar wave may be used for any mode of vibrations of the emitter and for any shape of the wall, but the computation becomes far easier if the wall is circular. It may be applied also for any wavelength of the radiated sound, but the functions expressing the dependence on the azimuthal angle and containing the Legendre associated functions of the first kind converge faster, the longer the wavelength in comparison with the dimensions of the wall. Numerical calculations are not given. They will be adequate for the difficulty of the problem, i.e., very tedious; but they can be done, especially with the help of modern electronic calculating machines.